Earning and Utility Limits in Fisher Markets.

Earning limits and utility limits are novel aspects in the classic Fisher market model. Sellers with earning limits have bounds on their income and lower the supply they bring to the market if income exceeds the limit. Buyers with utility limits have an upper bound on the amount of utility that they want to derive and lower the budget they bring to the market if utility exceeds the limit. Markets with these properties can have multiple equilibria with different characteristics. We analyze earning limits and utility limits in markets with linear and spending-constraint utilities. For markets with earning limits and spending-constraint utilities, we show that equilibrium price vectors form a lattice and the spending of buyers is unique in non-degenerate markets. We provide a scaling-based algorithm to compute an equilibrium in time O(n3ℓ log (ℓ + nU)), where n is the number of agents, ℓ ≥ n a bound on the segments in the utility functions, and U the largest integer in the market representation. We show how to refine any equilibrium in polynomial time to one with minimal prices or one with maximal prices (if it exists). Moreover, our algorithm can be used to obtain in polynomial time a 2-approximation for maximizing Nash social welfare in multi-unit markets with indivisible items that come in multiple copies. For markets with utility limits and linear utilities, we show similar results—lattice structure of price vectors, uniqueness of allocation in non-degenerate markets, and polynomial-time refinement procedures to obtain equilibria with minimal and maximal prices. We complement these positive results with hardness results for related computational questions. We prove that it is NP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.